Find the value of \int_{-\pi}^{\pi} (4 \arctan(e^x)-\pi)dx I've tried to show

Jason Yuhas

Jason Yuhas

Answered question


Find the value of ππ(4arctan(ex)π)dx

Answer & Explanation

Mary Goodson

Mary Goodson

Beginner2022-01-01Added 37 answers

Let f(x)=4arctan(ex)π. Then since arctanu+arctan1u=π2 (∗):
(∗) can be proved geometrically: arctanu is the angle with opposite side u and adjacent side 1, and arctan1u is the other acute angle in the right triangle. Their sum must thus be π2. Alternatively, differentiate the left-hand side and show it is constant, then choose a convenient value for u since any u works, say u=1.


Beginner2022-01-02Added 31 answers

Write the integral as
Upon making the substitution u=−x, we find that
Note that
arctan(u+arctan(1u))={π2 if u>0π2 if u<0
In this case, x[0,π], so u=ex>0 and arctan(u)+arctan(1u)=π2, meaning that the integrand is the zero function. Hence, I=0.


Expert2022-01-08Added 613 answers

The answer expands on my comment, which remarked that the integrand of the given integral is twice the Gudermannian function, gd, which appears most famously in the equation governing the Mercator projection in cartography.
Differentiating the integrand gives
In particular, this derivative is even. Since evaluating the integrand at x=0 gives 4arctane0π=0 the integrand is odd; since the integral is taken over an interval symmetric around 0, by symmetry
for any a: In particular, the occurrence of π in the limits of the integral is something of a red herring.

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